and apply plane-wave expansion of the exponential function,
Eq. (REF ). Substituting Eq. (REF ), the product of
second-order propagators in Eq. (REF ) is given by a product
of invariant propagators and a product of bipolar spherical harmonics.
The product of bipolar spherical harmonics reduces to a single bipolar
spherical harmonics according to a formula
[1]}, [2]}
\(\left\lbrace Y_{l_1^{\prime }}(\hat{{k}}^{\prime })\otimes Y_{l_1^{\prime \prime }}(\hat{{k}}^{\prime \prime })\right\rbrace _{l_1m_1}\left\lbrace Y_{l_2^{\prime }}(\hat{{k}}^{\prime })\otimes Y_{l_2^{\prime \prime }}(\hat{{k}}^{\prime \prime })\right\rbrace _{l_2m_2}\\=(-1)^{l_1+l_2}\frac{\sqrt{\lbrace l_1\rbrace \lbrace l_2\rbrace \lbrace l_1^{\prime }\rbrace \lbrace l_2^{\prime }\rbrace \lbrace l_1^{\prime \prime }\rbrace \lbrace l_2^{\prime \prime }\rbrace }}{4\pi }\sum _l \sqrt{\lbrace l\rbrace }\left(l\,l_1\,l_2\right)^m_{\phantom{m}m_1m_2}\\ \times \sum _{l^{\prime },l^{\prime \prime }} (-1)^{l^{\prime }+l^{\prime \prime }} \sqrt{\lbrace l^{\prime }\rbrace \lbrace l^{\prime \prime }\rbrace }\begin{pmatrix}l_1^{\prime } & l_2^{\prime } & l^{\prime } \\ 0 & 0 & 0\end{pmatrix}\begin{pmatrix}l_1^{\prime \prime } & l_2^{\prime \prime } & l^{\prime \prime } \\ 0 & 0 & 0\end{pmatrix}\\ \times \begin{Bmatrix}l_1 & l_2 & l \\l_1^{\prime } & l_2^{\prime } & l^{\prime } \\l_1^{\prime \prime } & l_2^{\prime \prime } & l^{\prime \prime }\end{Bmatrix}\left\lbrace Y_{l^{\prime }}(\hat{{k}}^{\prime })\otimes Y_{l^{\prime \prime }}(\hat{{k}}^{\prime \prime })\right\rbrace _{lm}.\) 
